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Properties

Label 2-504-56.27-c1-0-34
Degree $2$
Conductor $504$
Sign $0.751 + 0.659i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.37 − 0.310i)2-s + (1.80 − 0.856i)4-s + 2.33·5-s + (−0.490 − 2.59i)7-s + (2.22 − 1.74i)8-s + (3.22 − 0.724i)10-s + 0.304·11-s − 5.46·13-s + (−1.48 − 3.43i)14-s + (2.53 − 3.09i)16-s + 6.37i·17-s − 0.840i·19-s + (4.21 − 1.99i)20-s + (0.420 − 0.0945i)22-s − 0.111i·23-s + ⋯
L(s)  = 1  + (0.975 − 0.219i)2-s + (0.903 − 0.428i)4-s + 1.04·5-s + (−0.185 − 0.982i)7-s + (0.787 − 0.616i)8-s + (1.01 − 0.229i)10-s + 0.0918·11-s − 1.51·13-s + (−0.396 − 0.917i)14-s + (0.632 − 0.774i)16-s + 1.54i·17-s − 0.192i·19-s + (0.943 − 0.447i)20-s + (0.0895 − 0.0201i)22-s − 0.0232i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.751 + 0.659i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.751 + 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.68734 - 1.01198i\)
\(L(\frac12)\) \(\approx\) \(2.68734 - 1.01198i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 + 0.310i)T \)
3 \( 1 \)
7 \( 1 + (0.490 + 2.59i)T \)
good5 \( 1 - 2.33T + 5T^{2} \)
11 \( 1 - 0.304T + 11T^{2} \)
13 \( 1 + 5.46T + 13T^{2} \)
17 \( 1 - 6.37iT - 17T^{2} \)
19 \( 1 + 0.840iT - 19T^{2} \)
23 \( 1 + 0.111iT - 23T^{2} \)
29 \( 1 - 5.46iT - 29T^{2} \)
31 \( 1 - 7.64T + 31T^{2} \)
37 \( 1 - 6.44iT - 37T^{2} \)
41 \( 1 + 8.66iT - 41T^{2} \)
43 \( 1 - 6.06T + 43T^{2} \)
47 \( 1 + 8.21T + 47T^{2} \)
53 \( 1 - 10.1iT - 53T^{2} \)
59 \( 1 - 1.70iT - 59T^{2} \)
61 \( 1 - 2.75T + 61T^{2} \)
67 \( 1 + 8.35T + 67T^{2} \)
71 \( 1 - 2.07iT - 71T^{2} \)
73 \( 1 + 12.7iT - 73T^{2} \)
79 \( 1 + 7.90iT - 79T^{2} \)
83 \( 1 - 11.6iT - 83T^{2} \)
89 \( 1 + 4.08iT - 89T^{2} \)
97 \( 1 - 6.42iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.56669077924682205682739615550, −10.30542637043016530428759396187, −9.380843237007658944044751064897, −7.84784488933447770075066258502, −6.86546989771957902303230761323, −6.12079502581099064878848503112, −5.05765614217560694918550915538, −4.12795329025191119274432086110, −2.82395191187826537060798324811, −1.57751899988079067708256700217, 2.21167651979379249325490818229, 2.86169371809032666061341249342, 4.61958095808289769177474485790, 5.38200161482403765395241962714, 6.18857096826261897972763998742, 7.11283729333490145828090635502, 8.170314393215835481137563722004, 9.544860967269919891541151074301, 9.906809614902216217471704534294, 11.40098844001736971290675885865

Graph of the $Z$-function along the critical line